I’m old enough to remember encountering this phrase and thinking of it as a cute novelty. It’s a useful term, but it’s sort of meta that we didn’t have a way to describe something being a thing before “being a thing” became a thing. 
Or did we?
I’m having a hard time doing any searches on its true origin. (Turns out googling “thing” goes … places.)
Late 1990s to early 2000s, apparently Language Log » When did "a thing" become a thing?
Letterman had a kind of sideshow segment he did occasionally called “Is This A Thing?”. The curtain would go up and someone would be doing some unusual act. Dave and Paul would discuss whether it was a thing, in the sense of entertainment.
The big stars that were so well-liked that they were incorporated into every time the segment ran were Grinder Girl and Hula Hoop Girl.
Isn’t “being a thing” a sort of general term, and previously you would have used a more specific one?
At about the same time that the language of philosophy shifted from Latin to English. Before that, you’d speak of something being reified.
Back in the early 80’s I dated a lot of models. They referred to other models as “thing”.
Here is the earliest recorded usage in the OED:
2000 A. Sorkin West Wing (transcribed from TV programme) 2nd Ser. Episode 5 Did you know ‘leaf peeping’ was a thing?
There’s such a thing as purplehorseshoe.
purplehorseshoe exists.
I was going to post that I personally first remember hearing this on The West Wing.
I believe that usage that the OP is asking about is when people call a potential problem as “a thing”.
For example: “I’m thinking about transferring Bob to your department. I know the two of you used to date. Is this going to be a thing?”
As to what things were called before they were things I’d suggest zeitgeist. Not the original meaning of the term (as with meme), but the common modern usage is approximately the same meaning as in “X is a thing now”.
Not necessarily a problem, just a phenomenon.
As Inner Stickler’s quote shows, not necessarily a problem. For your example the answer to the OP is “before they were called problems”, for Inner Stickler’s “an activity people do”.
Or, as I wrote, “a thing” is a generic expression, and before it became a thing, people would use a more specific one.
“I keep hearing people say ‘is this a thing’, has ‘a thing’ become a generic idiom for any kind of phenomenon or issue?”
The relevant sense from the OED is:
“A genuine phenomenon, established practice, or discernible trend. Often in questions conveying surprise or incredulity (as is that (even) a thing?), or as an assertion, esp. responding to or pre-empting scepticism (as it’s a thing).”
There are several related senses in which “a thing” is used.
One of them used to be called a “fad.”
“When did jumping out of a moving car and dancing become a [thing | fad]?”
A thing is something like a phenomenon.
I suspect the phrase is likely to be a development of “thing is” constructions, which are used to explain that a particular problem or outcome is a consequence of a more generally prevailing circumstance or condition. (“I’d love to come to Paris with you but the thing is, I don’t have any money.”) Thing is constructions are long established in colloquial speech, but rare in writing. “Thing” here refers to an antecedent condition, something that already prevails, but we only have to take note of now. So “when did X become a thing?” means “when did X become so commonplace that we have to note it, take account of it or acknowledge it?”
Before there were things, there were deals, as in, “what’s the deal with X?”
And when a deal became popular, it turned into a Big Deal.
1972 when Volkswagen introduced “The Thing” in the U.S.
I remember in the late 60’s/early 70’s, “That ain’t my thing” or “That’s my thing” being used before “bag” replaced thing. “That ain’t my bag, man.” or “That’s my bag.”.